Fonctions trigonométriques
\[\begin{align*} x/\sin x &= 1 + x^2/6 + 7x^4/360 + o(x^4) \cr 1/\cos x &= 1 + x^2/2 + 5x^4/24 + o(x^4) \cr \ln(\sin x/x) &= -x^2/6 -x^4/180 -x^6/2835 + o(x^6) \cr \exp(\sin x/x) &= e(1 - x^2/6 + x^4/45) + o(x^4) \cr \sqrt{\tan x} &= 1 + h + h^2/2 + o(h^2), h = x-\pi/4 \cr \sin(x+x^2+x^3-x^4) &= x + x^2 + 5x^3/6 -3x^4/2 + o(x^4) \cr \ln(x\tan(1/x)) &= x^{-2}/3 + 7x^{-4}/90 + o(1/x^4) \cr (1-\cos x)/(e^x-1)^2 &= 1/2 - x/2 + x^2/6 + o(x^2) \cr \sin((\pi\cos x)/2) &=1 -\pi^2x^4/32 + \pi^2x^6/192 + o(x^6) \cr \cos x\ln(1+x) &= x - x^2/2 - x^3/6 + o(x^4) \cr (\sin x-1)/(\cos x+1) &= -1/2 + x/2 - x^2/8 + o(x^2) \cr \ln(2\cos x+\tan x) &= \ln2+x/2-5x^2/8+11x^3/24-59x^4/192 + o(x^4) \cr e^{\cos x} &= e(1 - x^2/2 + x^4/6) + o(x^5) \cr \end{align*}\]
Fonctions circulaires inverses
\[\begin{align*} \arcsin^2 x &= x^2 + x^4/3 + 8x^6/45 + o(x^6) \cr 1/\arcsin^2 x &= x^{-2} - 1/3 - x^2/15 + o(x^2) \cr \arctan\sqrt{(x+1)/(x+2)} &= \pi/4 - x^{-1}/4 + 3x^{-2}/8 \cr \arccos(\sin x/x) &= |x|/\sqrt3(1 - x^2/90) + o(1/x^3) \cr 1/\arctan x &= x^{-1} + x/3 -4x^3/45 +44x^5/945 + o(x^5) \cr \arcsin\sqrt x &= \pi/6+1/\sqrt3(2h-4h^2/3+32h^3/9) + o(h^3), h = x-1/4 \cr \arcsin(\sin^2 x) &= x^2 -x^4/3 +19x^6/90 -107x^8/630 + o(x^8) \cr \arctan(1+x) &= \pi/4 + x/2 - x^2/4 + x^3/12 + o(x^4) \cr \arcsin x/(x-x^2) &= 1 + x + 7x^2/6 + o(x^2) \cr e^{\arcsin x} &= e^{\pi/6}(1 + 2h/\sqrt3 + 2(1+\sqrt3)h^2/(3\sqrt3)) + o(h^2), h = x-1/2 \cr e^{1/x}\arctan x &= \frac\pi2+ (\frac\pi2-1)x^{-1} + (\frac\pi4-1)x^{-2} + (\frac\pi{12}-\frac16)x^{-3} + o(1/x^3) \cr \end{align*}\]
Exponentielle et logarithme
\[\begin{align*} x/(e^x-1) &= 1 - x/2 + x^2/12 + o(x^2) \cr \ln x/\sqrt x &= h - h^2 + 23h^3/24 + o(h^3), h = x-1 \cr \ln((2-x)/(3-x^2)) &= \ln(2/3) - x/2 + 5x^2/24 + o(x^2) \cr \ln(1+x)/(1-x+x^2) &= x + x^2/2 - x^3/6 + o(x^3) \cr \ch x/\ln(1+x) &= x^{-1} + 1/2 + 5x/12 + o(x) \cr \ln(\ln(1+x)/x) &= -x/2 + 5x^2/24 - x^3/8 + o(x^3) \cr \ln(a^x+b^x) &= \ln2 + x\ln\sqrt{ab} + x^2\ln^2(a/b)/8 + o(x^2) \cr \exp(1/x)/x^2 &= e(1 - 3h + 13h^2/2 - 73h^3/6) + o(h^3), h = x-1 \cr \end{align*}\]
Fonctions hyperboliques inverses
\[\begin{align*} \Argth(\sin x) &= x + x^3/6 + x^5/24 + o(x^5) \cr \Argsh(e^x) &= \ln(1+\sqrt2) + 1/\sqrt2(x + x^2/4) + o(x^2) \cr \end{align*}\]
Formes exponentielles
\[\begin{align*} (1-x+x^2)^{1/x} &= e^{-1}(1 + x/2 + 19x^2/24) + o(x^2) \cr ((1+x)/(1-x))^\alpha &= 1 + 2\alpha x + 2\alpha^2x^2 + 2\alpha(2\alpha^2+1)x^3/3 + o(x^3) \cr (\sin x/x)^{2/x^2} &= e^{-1/3}(1 - x^2/90) + o(x^3) \cr (\sin x/x)^{3/x^2} &= e^{-1/2}(1 - x^2/60 - 139x^4/151200)+ o(x^4) \cr (1+\sin x)^{1/x} &= e(1 - x/2 + 7x^2/24) + o(x^2) \cr (1+\sin x + \cos x)^x &= 1 + x\ln2 + x^2(\ln^22+1)/2 + o(x^2) \cr (\sin x)^{\sin x} &= 1 - h^2/2 + 7h^4/24 + o(h^4), h = x-\pi/2 \cr (\tan x)^{\tan2x} &= e^{-1}(1 + 2h^2/3 + 4h^4/5) + o(h^4), h = x-\pi/4 \cr & \text{Développer d'abord $\ln((1+x)/(1-x))$} \cr \end{align*}\]
Radicaux
\[\begin{align*} x\sqrt{(x-1)/(x+1)} &= 1/\sqrt3(2 + 5h/3 + h^3/54) + o(h^3), h = x-2 \cr \sqrt{1+\sqrt{1-x}} &= \sqrt2(1 - x/8 - 5x^2/128 - 21x^3/1024) + o(x^3) \cr \sqrt{1-\sqrt{1-x^2}} &= |x|/\sqrt2(1 + x^2/8 + 7x^4/128) + o(x^5) \cr e^x-\sqrt{1+2x} &= x^2 - x^3/3 + 2x^4/3 - 13x^5/15 + o(x^5) \cr (\sqrt[3]{x^3+x^2}+\sqrt[3]{x^3-x^2})/x &= 2 - 2x^{-2}/9 + o(1/x^3) \cr \end{align*}\]